Theorem cbvrmow 2691

Description: Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Version of cbvrmo 2741 with a disjoint variable condition. (Contributed by NM, 16-Jun-2017.) (Revised by GG, 23-May-2024.)

Hypotheses

Ref	Expression
cbvrmow.1	|- F/ y ph
cbvrmow.2	|- F/ x ps
cbvrmow.3	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvrmow	|- ( E* x e. A ph <-> E* y e. A ps )

Distinct variable group:   x, y, A

Allowed substitution hints:   ph( x, y)   ps( x, y)

Proof of Theorem cbvrmow

Step	Hyp	Ref	Expression
1		nfv 1552	. . . 4 |- F/ y x e. A
2		cbvrmow.1	. . . 4 |- F/ y ph
3	1, 2	nfan 1589	. . 3 |- F/ y ( x e. A /\ ph )
4		nfv 1552	. . . 4 |- F/ x y e. A
5		cbvrmow.2	. . . 4 |- F/ x ps
6	4, 5	nfan 1589	. . 3 |- F/ x ( y e. A /\ ps )
7		eleq1w 2268	. . . 4 |- ( x = y -> ( x e. A <-> y e. A ) )
8		cbvrmow.3	. . . 4 |- ( x = y -> ( ph <-> ps ) )
9	7, 8	anbi12d 473	. . 3 |- ( x = y -> ( ( x e. A /\ ph ) <-> ( y e. A /\ ps ) ) )
10	3, 6, 9	cbvmow 2096	. 2 |- ( E* x ( x e. A /\ ph ) <-> E* y ( y e. A /\ ps ) )
11		df-rmo 2494	. 2 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
12		df-rmo 2494	. 2 |- ( E* y e. A ps <-> E* y ( y e. A /\ ps ) )
13	10, 11, 12	3bitr4i 212	1 |- ( E* x e. A ph <-> E* y e. A ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   F/wnf 1484   E*wmo 2056   e. wcel 2178   E*wrmo 2489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clel 2203  df-rmo 2494

This theorem is referenced by:  cbvreuw  2737